Tolkien was playing coy with this story for several reasons. One, it would be too easy to make it recognizable as 'oh, hey - that's just the Garden of Eden story.' But more importantly, he thought it a mistake to introduce the actual concept of Christianity into a fantasy world. And thirdly...these stories are elf-centric. So, the tale of the Fall of Man is not really something they had anything to do with and not something they 'get'.
He wanted his mythology to not explicitly include the Christian religion, but to be believable/recognizable to someone who believed what a Christian believes. Obviously, the line is crossed when you introduce the Incarnation of Eru to bring about the redemption of mankind. (An idea that comes up in the Athrabeth.) And, from a Christian perspective, that need for redemption finds its origin in...the Fall.
Original Man (Unfallen Man) has a tendency to act according to God's will. He has freedom of choice, but his natural inclinations are aligned with...being a good person. So, it's 'natural' to be good and do the right thing. This person does not experience temptation as an 'urge' to do something that they want to, but know they shouldn't. Therefore, temptation is more of an intellectual exercise rather than an issue of enticement via more carnal means.
Then the Fall happens, and oh look, people don't always follow their consciences. Concupiscence exists, and temptation is now something to be struggled with. What you want to do and ought to do isn't necessarily what you wind up doing, because there were urges to do something else that overrode your good intentions. The struggle is real, and lots of horrible evil things are done. Humanity is a mess.
And then, REDEMPTION. What Jesus offers was (among other things) a life perfectly lived. Through grace, there can be a purification of the heart so that what we want is again aligned with what is good and the will of God. There is no complete overcoming of concupiscence in this life time (even saints have their faults), but this purification continues into heaven, resulting in a new Redeemed Man - similar to Original Man, but with the lived experience of Fallen Man.
Pretty much all we see is Fallen Man, with glimpses of Redeemed Man here and there, but we've never seen Original Man, and don't know what it would be like to live in a society where all people were fully redeemed. So....it's an interesting way of bookending human experience, and certainly something Tolkien was very cognizant of...but how much of this do we want to introduce into our project? The idea of an original sin that stains human nature so that it can never again be as it was created to be without direct intervention by God...well, that's just 'default', really. By the time we meet Men, they will be Fallen, which is to say...normal human beings. The real challenge is to show the elves as UNFALLEN, when the Silmarillion elves spend so much time doing all the awful unimaginable things that fallen Men do.
So what do we show? We will have a glimpse of the newly awakened Men in a paradise-like setting, but...that's really our only allusion to their origins. We aren't going to show enough story with them to show them acting or behaving in any particular way as Unfallen creatures. What can we have be their reason for falling? Worshipping Morgoth would certainly do the trick, but...why would they? What could Morgoth (or Sauron) say or do to convince them that what they really want out of life is to follow the dark god? Do we want to convey some other temptation?